How Are Hamiltonian and Lagrangian Related in Quantum Mechanics?

[TimeRip496]

"The hamiltonian runs over the time axis while the lagrangian runs over the trajectory of the moving particle, the t'-axis."
What does the above statement means? Isnt hamiltonian just an operator that corresponds to total energy of a system? How is hamiltonian related to lagrangian intuitively?

Besides what is lagrangian density intuitively and mathematically? Is it equal to lagrangian?

 

[robphy]

Is this related to the source of your quote?

http://physics.stackexchange.com/qu...meaning-of-the-action-in-lagrangian-mechanics

 

[TimeRip496]

Is this related to the source of your quote?

http://physics.stackexchange.com/qu...meaning-of-the-action-in-lagrangian-mechanics

Yes. But I don't really understand it so I post it here.

 

[robphy]

In the future, it's best to provide a reference to a quote so that folks can easily look at the source and context of the quote.
I had to do a Google search and guessed this was the source.

Concerning the statement of the quote...
The interpretation doesn't seem to be a standard interpretation... but it seems interesting and might be worthy of further study.
I can't say I understand enough of that interpretation to give a summary of the idea. (Do a search for ... hans de vries largrangian ).

Since you refer to the Hamiltonian as an operator, your context seems to be quantum mechanics or quantum field theory, which appears to be the context of the quote. (See Ch 22 of the document that comes up in the Google search.)

In classical mechanics, the Hamiltonian and Lagrangian are related by a Legendre Transformation.
I don't have yet an "intuitive" explanation of that relationship... beyond saying it's an important transformation of variables. (part of a backburner project)

While the largrangian is used in particle mechanics (with few degrees of freedom),
the Lagrangian density is used in field theory (with many more degrees of freedom).
Rather than being a function of configurations and velocities,
it is a function of the field values and their derivatives in some region of space.
Crudely speaking, the Lagrangian density is in some sense the Lagrangian-per-unit-volume.

Possibly useful:
http://en.wikipedia.org/wiki/Lagrangian